Abstract
We introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-14-CE27-0019
Award Identifier / Grant number: ANR-12-BS01-0014
Award Identifier / Grant number: ANR-10-LABX-0070
Award Identifier / Grant number: ANR-11-IDEX-0007
Funding statement: Simon Masnou and François Dayrens acknowledge support from the French National Research Agency (ANR) research grants MIRIAM (ANR-14-CE27-0019) and GEOMETRYA (ANR-12-BS01-0014), and from the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007). Matteo Novaga and Marco Pozzetta acknowledge support from the INdAM-GNAMPA Project 2019 Problemi geometrici per strutture singolari.
References
[1] L. Ambrosio, V. Caselles, S. Masnou and J.-M. Morel, Connected components of sets of finite perimeter and applications to image processing, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 1, 39–92. 10.1007/PL00011302Search in Google Scholar
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University, New York, 2000. Search in Google Scholar
[3] M. Bonnivard, A. Lemenant and V. Millot, On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers, Interfaces Free Bound. 20 (2018), no. 1, 69–106. 10.4171/IFB/397Search in Google Scholar
[4] M. Bonnivard, A. Lemenant and F. Santambrogio, Approximation of length minimization problems among compact connected sets, SIAM J. Math. Anal. 47 (2015), no. 2, 1489–1529. 10.1137/14096061XSearch in Google Scholar
[5] P. W. Dondl, A. Lemenant and S. Wojtowytsch, Phase field models for thin elastic structures with topological constraint, Arch. Ration. Mech. Anal. 223 (2017), no. 2, 693–736. 10.1007/s00205-016-1043-6Search in Google Scholar
[6] P. W. Dondl, L. Mugnai and M. Röger, A phase field model for the optimization of the Willmore energy in the class of connected surfaces, SIAM J. Math. Anal. 46 (2014), no. 2, 1610–1632. 10.1137/130921994Search in Google Scholar
[7] P. W. Dondl, M. Novaga, B. Wirth and S. Wojtowytsch, A phase-field approximation of the perimeter under a connectedness constraint, SIAM J. Math. Anal. 51 (2019), no. 5, 3902–3920. 10.1137/18M1225197Search in Google Scholar
[8] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969. Search in Google Scholar
[9] G. Gamow, Mass defect curve and nuclear constitution, Proc. Roy. Soc. Lond. A 126 (1930), 632–644. 10.1098/rspa.1930.0032Search in Google Scholar
[10] E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16 (1968), 1–29. 10.1137/0116001Search in Google Scholar
[11] M. Goldman, M. Novaga and B. Ruffini, Existence and stability for a non-local isoperimetric model of charged liquid drops, Arch. Ration. Mech. Anal. 217 (2015), no. 1, 1–36. 10.1007/s00205-014-0827-9Search in Google Scholar
[12] H. Knüpfer and C. B. Muratov, On an isoperimetric problem with a competing nonlocal term I: The planar case, Comm. Pure Appl. Math. 66 (2013), no. 7, 1129–1162. 10.1002/cpa.21451Search in Google Scholar
[13] C. B. Muratov, M. Novaga and B. Ruffini, On equilibrium shape of charged flat drops, Comm. Pure Appl. Math. 71 (2018), no. 6, 1049–1073. 10.1002/cpa.21739Search in Google Scholar
[14] E. Paolini and E. Stepanov, Existence and regularity results for the Steiner problem, Calc. Var. Partial Differential Equations 46 (2013), no. 3–4, 837–860. 10.1007/s00526-012-0505-4Search in Google Scholar
[15] M. Pozzetta, A varifold persepctive on the p-elastic energy of planar sets, preprint (2019), https://arxiv.org/abs/1902.10463; to appear in J. Convex Anal. Search in Google Scholar
[16] T. Schmidt, Strict interior approximation of sets of finite perimeter and functions of bounded variation, Proc. Amer. Math. Soc. 143 (2015), no. 5, 2069–2084. 10.1090/S0002-9939-2014-12381-1Search in Google Scholar
[17] U. Seifert, Configurations of fluid membranes and vesicles, Adv. Phys. 46 (1997), no. 1, 13–137. 10.1080/00018739700101488Search in Google Scholar
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